Proof of Proposition 4: We first show that the dynamics are the sum over endowment and noise shocks of the dynamics generated by each shock. We then determine the dynamics generated by an endowment shock.
We define the vectors v`−1 and b by
Equation (A21) implies that in equilibrium
v` = N v`−1+ nu`+ b`+1. Iterating this equation between periods m0 and m0, we get
vm0 = Nm0−m0vm0 +
The first term in equation (A35) corresponds to the dynamics generated by the endowment and noise shocks prior to period m0. The second and third terms correspond to the dynamics generated by the shocks between periods m0 and m0.
To isolate the dynamics generated by a unit endowment shock in period `, we use equation (A35) and set m0 = `0− 1, vm0 = 0, um = 0 for all m, m = 0 for all m 6= `, and
` = 1. We get
v`0−1= N`0−`b. (A36)
To determine v`0−1, we need to determine the eigenvalues and eigenvectors of the matrix N , and write b as a linear combination of the eigenvectors. The eigenvalues of N are 1, 1 − as− ae, and 1 − ae(1 + g), and the corresponding eigenvectors are Equation (30) follows from equation (A37), by noting that e`0−1 is the sum of the first and second components of v`0−1. The first component of v`0−1 is e`0−1− s`0−1 and goes to 0.
we get equation (31). Q.E.D.
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0 1
0 20 40 60 80 100
Trading Days
Stock Holdings and Price
Stock Holdings Price
Long-Run Limit
Figure 1. The first convergence pattern. The thick solid line represents the large trader’s stock holdings, the dotted line the price, and the thin solid line the long-run limit of stock holdings and price. (Price is normalized so that it has the same initial value and long-run limit as stock holdings.) The large trader’s stock holdings decrease over time. The trading rate, which is the ratio of the slope of stock holdings to the difference between stock holdings and the long-run limit, decreases over time. The price impact, which is the ratio of the slope of price to the slope of stock holdings, also decreases over time. This figure is drawn for the following parameter values. The parameter σ2u = σu2/σe2 is equal to 2, which means that the small traders produce twice as much order flow as the large trader. The parameter α/α is equal to 1/9, which means that the large trader is nine times as risk-averse as the market makers as a group. The annual interest rate r, is equal to 5 percent. The time between trading periods h, is equal to 1/5000’th of a year. Since there are approximately 250 trading days per year, our choice of h implies 20 trading periods per day. Finally, the parameter ασσe/r is equal to 0.1. This parameter determines the ratio of price risk (price variability due to trades) to dividend risk (price variability due to dividends). Our choice of ασσe/r implies a ratio of approximately 10 percent.
0 1
0 20 40 60 80 100
Trading Days
Stock Holdings and Price
Stock Holdings Price
Long-Run Limit
Figure 2. The second convergence pattern. The thick solid line represents the large trader’s stock holdings, the dotted line the price, and the thin solid line the long-run limit of stock holdings and price. The large trader’s stock holdings decrease and then increase over time. First, they decrease to their long-run limit. Then, they decrease further, and increase back to their long-run limit, as the large trader engages in a “round-trip transaction”.
Notice that the round-trip transaction is profitable, since the average price of sales exceeds the average price of purchases. The parameter values for this figure are σ2u = 8 (the small traders produce eight times as much order flow as the large trader), α/α = 1/9 (the large trader is nine times as risk-averse as the market makers as a group), r = 5 percent, h = 1/5000’th of a year (20 trading periods per day), and ασσe/r = 0.1.
0 1
0 20 40 60 80 100
Trading Days
Stock Holdings
h1 h2<h1
Long-Run Limit
Figure 3. The dynamics of the large trader’s stock holdings for small time between trades h. The thick solid line represents stock holdings for h1, the dotted line stock holdings for h2< h1, and the thin solid line the long-run limit of stock holdings. The dynamics consist of two phases. In the first, short phase, stock holdings decrease quickly, while in the second, long phase, they evolve more slowly. Notice that the first phase is shorter for h2 than for h1. The parameter values for this figure are σ2u = 2 (the small traders produce twice as much order flow as the large trader), α/α = 1/9 (the large trader is nine times as risk-averse as the market makers as a group), r = 5 percent, h1 = 1/5000 and h2= 1/50000’th of a year (20 or 200 trading periods per day), and ασσe/r = 0.1.
0 1
0 20 40 60 80 100
Trading Days
Stock Holdings
Noise 1
Noise 2 > Noise 1 Noise 3 > Noise 2 Long-Run Limit
Figure 4. The effects of the noise σ2u, on the dynamics of the large trader’s stock holdings. The thick solid line represents stock holdings for (σ2u)1, the thick dotted line stock holdings for (σ2u)2 > (σ2u)1, the thin dotted line stock holdings for (σ2u)3 > (σ2u)2, and the thin solid line the long-run limit of stock holdings. As the noise increases, the initial drop in stock holdings increases, and the large trader sells faster. For (σ2u)3, stock holdings drop below their long-run limit, and thus decrease and then increase over time.
The parameter values for this figure are (σ2u)1 = 2, (σ2u)2 = 4, and (σ2u)3 = 8 (the small traders produce two, four, or eight times as much order flow as the large trader), α/α = 1/9 (the large trader is nine times as risk-averse as the market makers as a group), r = 5 percent, h = 1/5000’th of a year (20 trading periods per day), and ασσe/r = 0.1.
0 1
0 20 40 60 80 100
Trading Days
Stock Holdings
Risk Aversion 1 Long-Run Limit 1
Risk Aversion 2 < Risk Aversion 1 Long-Run Limit 2
Figure 5. The effects of the large trader’s risk aversion α, on the dynamics of the large trader’s stock holdings. The thick solid line represents stock holdings for α1, the thick dotted line stock holdings for α2 < α1, the thin solid line the long-run limit of stock holdings for α1, and the thin dotted line the long-run limit for α2. As α decreases, the long-run limit of stock holdings increases, while the initial drop stays roughly the same.
For α2, stock holdings drop below their long-run limit, and thus decrease and then increase over time. Since for α2 stock holdings drop below their long-run limit, the large trader sells faster for α2 than for α1. The parameter values for this figure are σ2u = 2 (the small traders produce twice as much order flow as the large trader), α/α1= 1/6 and α/α2 = 1/4 (the large trader is six or four times as risk-averse as the market makers as a group), r = 5 percent, h = 1/5000’th of a year (20 trading periods per day), and ασσe/r = 0.1.
0 25 50 75 100
0 2 4 6 8 10 12 14 16 18 20
Noise
ATTE (Trading Days)
Figure 6. The average time of trade execution as a function of the noise. The parameter values for this figure are α/α = 1/9 (the large trader is nine times as risk-averse as the market makers as a group), r = 5 percent, h = 1/5000’th of a year (20 trading periods per day), and ασσe/r = 0.1.
Notes
1Schwartz and Shapiro (1992) estimate that in 1989 about 70 percent of the trading volume in the NYSE was accounted for by member firms and institutional investors.
2Holthausen, Leftwich, and Mayers (1990) examine the 50 largest buy and sell trades for 109 randomly selected NYSE firms in 1983. They find a price impact of around 1 percent.
Keim and Madhavan (1996) examine block trades for small NYSE, AMEX, and Nasdaq firms, from 1985 to 1992, and find a much larger price impact of around 8 percent. See also Kraus and Stoll (1972), Scholes (1972), Holthausen, Leftwich, and Mayers (1987), Hausman, Lo, and McKinlay (1992), and Chan and Lakonishok (1993).
3Chan and Lakonishok (1995) examine the trades of 37 large investment management firms from 1986 to 1988. They find that individual trades are generally part of a sequence or, in their terminology, a “package”, and that 53 percent of packages take four or more days to be completed.
4Kyle’s paper has been extended in a number of directions. Holden and Subrahmanyam (1992) assume many insiders who have the same information. Foster and Vishwanathan (1996) and Back, Cao, and Willard (2000) assume many insiders who have different pieces of information. Back (1992) assumes general distributions for asset payoffs. Gennotte and Kyle (1993) assume an infinite horizon. Holden and Subrahmanyam (1994) and Baruch (1999) assume risk-averse insiders. See also the survey by Admati (1991) and the book by O’Hara (1995).
5For evidence on the performance of mutual and pension funds, see the survey by Fama (1991). For evidence on turnover, see, for instance, Chevalier and Ellison (1999). They report that the growth, and growth and income funds in the 1994 Morningstar CD turn over 76.8 percent of their portfolios every year, but underperform the market by 0.5 percent.
6We choose risk-sharing over portfolio rebalancing and liquidity because it is easier to model. Portfolio rebalancing requires wealth effects and a different utility function than the exponential. Liquidity requires borrowing constraints.
7The stock endowment can be interpreted as a futures position or as a position in a correlated stock.
8The assumption that market makers are risk-averse was standard in the early, “inventory-based”, market-microstructure literature. For a survey of that literature and for more recent references, see O’Hara (1995).
9We explain why the results are different in Section IV.B.
10The time between trades h, is smaller than a day. Therefore, a criticism of the utility in equation (2) is that it connects intraday trading decisions and consumption/savings decisions. To address this criticism, we could assume that consumption takes place once every given number of periods. (Traders are evaluated every quarter.) This assumption would complicate the model but would probably not change the results.
11We introduce the consumption good endowment for tractability. In our model, the large trader faces “fundamental” risk (the risk of a negative dividend shock), “price” risk (the risk of selling at the same time as the small traders), and “endowment” risk (the risk of receiving a small stock endowment). The latter risk depends on the large trader’s marginal valuation of a share. Therefore, it increases in the dividend level and decreases in the large trader’s stock holdings. The large trader can hedge this risk by increasing his stock holdings, i.e. buying shares from the market makers. Moreover, he needs to buy more shares when dividends are high. The consumption good endowment makes the hedging demand independent of the dividend level, thus simplifying the model. Indeed, the consumption good endowment represents a cost that the large trader pays in order to receive the stock endowment. The risk of receiving a small stock endowment is accompanied by the risk of paying a small cost. The specification of the consumption good endowment ensures that the sum of the two risks is independent of the dividend level. The model without the consumption good endowment was considered in an earlier version and produces similar results. Moreover, the model with the consumption good endowment is, in a sense, more realistic, since it captures the idea that the stock endowment does not come for “free”. If, for instance, the stock endowment is interpreted as a position in a correlated stock, the consumption good endowment captures the cost of establishing the position.
12Notice that the large trader is treated as an “atom”, while a market maker is treated as part of a continuum. It is in this sense that the large trader is large, and thus strategic, while a market maker is small (infinitesimal), and thus competitive. Notice also that a continuum of market makers is equivalent to a single competitive market maker. We assume
a continuum because this clarifies the exposition.
13In this paper we determine one stationary Nash equilibrium, and thus show existence of Nash equilibrium. We do not show uniqueness.
14The large trader’s sell order depends on the market makers’ expectation s`−1, and the market makers’ stock holdings e`−1. The large trader does not observe these variables directly. However, he can infer them from the prices up to period ` − 1. Indeed, equations (7), (10), and (15), imply that s` and e` can be obtained recursively from s`−1, e`−1, d`, and p`.
15 We thus assume that the variance of e`−1 conditional on I`−1 is independent of `.
This is a condition for stationarity and is satisfied if and only if Σ2e is equal to the unique positive root of equation
Σ2e= (1 − ae)2(Σ2e+ σe2h)σu2h
a2e(Σ2e+ σ2eh) + σu2h . (14) Indeed, the regression of e` on x` + u` implies that the variance of e` conditional on I` is equal to the RHS of equation (14). Notice that in our stationary equilibrium, Σ2e is endogenous since it is given by equation (14). The endogeneity of Σ2e raises an important issue. Suppose that an exogenous parameter, say σ2u, changes from (σ2u)1 to (σu2)2. Then Σ2e must also change, say from (Σ2e)1 to (Σ2e)2, since in our stationary equilibrium it is endogenous. Suppose now that the economy starts at a finite time instead of −∞. Then the initial value of Σ2e is an exogenous parameter. Moreover, a ceteris paribus comparison requires holding this initial value constant and equal to (Σ2e)1 when σ2u changes to (σu2)2. In this case, however, we obtain a non-stationary equilibrium and not the stationary equi-librium for (σu2)2. Focusing on the stationary equilibrium is valid, if it is the limit of the non-stationary equilibrium when ` goes to ∞. Indeed, in the limit, Σ2e is endogenous and equal to (Σ2e)2, and is independent of the initial value (Σ2e)1. We studied the non-stationary equilibrium assuming that the initial value (Σ2e)1 is close to the stationary equilibrium value (Σ2e)2 and showed (numerically) that convergence indeed obtains. This “local convergence”
result ensures that comparing stationary equilibria is valid, when the change in σu2 is small.
Indeed, the stationary equilibrium value of Σ2e is continuous in σu2, and thus (Σ2e)1 is close to (Σ2e)2 when (σ2u)1 is close to (σu2)2. By starting the economy at −∞ we side-step the above issue. We however implicitly assume that the economy converges to our stationary equilibrium.
16This is shown formally in the proof of Proposition 4.
17The risk-sharing rule is different in the presence of price risk. The large trader, who is subject to endowment shocks, is more reluctant to take risk than the market makers.
18When stock holdings decrease and then increase over time, the trading rate and the price impact have a complicated behavior, because their denominators can become 0.
19The parameter values for which this and the subsequent figures are drawn are in the figures’ legends. For now, we focus on the qualitative aspects of the figures. We discuss the parameter values and the axis markings in Section VII where we calibrate the model.
20Holden and Subrahmanyam (1994) and Baruch (1999) have shown that risk aversion makes the insider impatient. The insider prefers to establish a position early, before the noise traders introduce too much price volatility.
21It is worth noting that for a fixed h (i.e. not in the limit when h goes to 0) the large trader reveals his information more slowly when he is less risk-averse. This can be seen from Figure 5, where the first phase of the dynamics is longer when the large trader is less risk-averse. However, in the limit when h goes to 0, the large trader reveals his information immediately, independently of his risk aversion.
22Gennotte and Kyle (1993) and Chau (2000) study infinite-horizon models with insid-ers. However, Gennotte and Kyle do not report any results on the speed of information revelation, and Chau does not consider the continuous-time case.
23The proof is available from the author upon request. To show that φs+ φe increases in α and decreases in α, we need to make the plausible assumption that α < 2α, i.e. the market makers as a group are less than twice as risk-averse as the large trader.
24We focus on the effects of the noise, because it is the parameter that affects the ATTE the most.
25It is worth emphasizing that h affects the speed of trade only because it affects the length of the first phase. (This length is of order √
h.) Indeed, when h decreases, the number of trading periods increases, but the quantity traded per period decreases. In the first phase the first effect dominates, because the number of trading periods in a given time interval is of order 1/h, while the quantity traded per period is of order √
h. In the second
phase, however, the two effects cancel, because the quantity traded per period is of order h.
26The transversality condition (A6) is standard for optimal consumption-investment prob-lems. See, for instance, Merton (1969) and Wang (1994).